Cops and Attacking Robbers with Cycle Constraints

TL;DR

This paper studies a variant of the Cops and Robbers game where the robber can attack, characterizing certain graph classes and establishing bounds on the number of cops needed, with implications for graph theory and pursuit-evasion games.

Contribution

It characterizes triangle-free graphs with two cops, proves bounds for bipartite planar graphs, and constructs graphs demonstrating a significant difference between cop numbers in attacking and non-attacking versions.

Findings

Triangle-free graphs with two cops characterized

Bipartite planar graphs have at most four cops needed

Constructed graphs with a difference of three in cop numbers

Abstract

This paper considers the Cops and Attacking Robbers game, a variant of Cops and Robbers, where the robber is empowered to attack a cop in the same way a cop can capture the robber. In a graph $G$, the number of cops required to capture a robber in the Cops and Attacking Robbers game is denoted by $\attCop(G)$. We characterise the triangle-free graphs $G$ with $\attCop(G) \leq 2$ via a natural generalisation of the cop-win characterisation by Nowakowski and Winkler \cite{nowakowski1983vertex}. We also prove that all bipartite planar graphs $G$ have $\attCop(G) \leq 4$ and show this is tight by constructing a bipartite planar graph $G$ with $\attCop(G) = 4$. Finally we construct $17$ non-isomorphic graphs $H$ of order $58$ with $\attCop(H) = 6$ and $\cop(H)=3$. This provides the first example of a graph $H$ with $\attCop(H) - \cop(H) \geq 3$ extending work by Bonato, Finbow, Gordinowicz, Haidar, Kinnersley, Mitsche, Pra\l{}at, and Stacho \cite{bonato2014robber}. We conclude with a list of conjectures and open problems.